In city A, the streets are aligned in a grid, where the east

Let me add a little bit of detail to the solution given above.

You need to go from a point that is to the bottom left to a point that is to the top right. So you should take steps towards right and top. Since Bill wants to take shortest possible time, he should not go left or down because that is the opposite direction. His destination lies towards right and up.
Say, he takes one step to go from one intersection to the next one. He can take various routes e.g.
RRRRRUUUU (R represents one step right and U represents one step up)
RRRRUUUUR
etc

The total number of ways is basically obtained by re-arranging 5 Rs and 4 Us. You can do it in 9!/5!*4! = 126 ways (we divide by 5! and 4! because all Rs and all Us are identical)

Now, what happens due to the park? Everything is the same except that one intersection is not available - 4th Rd, 6th Ave. You cannot include this intersection in your journey. So what do you do? You remove all paths that include this intersection.

Now our question is this: In how many ways can you go from the corner of 2nd Rd and 3rd Ave to the corner of 6th Rd and 8th Ave when you include the corner of 4th Rd, 6th Ave?
From the corner of 2nd Rd and 3rd Ave to the corner of 4th Rd and 6th Ave - You need to take 3 steps right and 2 steps up - RRRUU etc. No of ways = 5!/3!*2! = 10
From the corner of 4th Rd and 6th Ave to the corner of 6th Rd and 8th Ave - You need to take 2 steps right and 2 steps up - RRUU etc. No of ways = 4!/2!*2! = 6

Total number of ways in which the 4th Rd, 6th Ave is included = 10*6 = 60 ways

Number of paths in which the corner of 4th Rd, 6th Ave is not included = 126 - 60 = 66